learning Outcomes

After reading this chapter, the reader will be able to

• Understand the meaning of three processes of heat flow: conduction, convection, and radiation

• Know about thermal conductivity, diffusivity, and steady-state condition of a thermal conductor

• Derive Fourier's one-dimensional heat flow equation and solve it in the steady state

• Derive the mathematical expression for the temperature distribution in a lagged bar

• Derive the amount of heat flow in a cylindrical and a spherical thermal conductor

• Solve numerical problems and multiple choice questions on the process of conduction of heat

6.1 Introduction

Heat is the thermal energy transferred between different substances that are maintained at different temperatures. This energy is always transferred from the hotter object (which is maintained at a higher temperature) to the colder one (which is maintained at a lower temperature). Heat is the energy arising due to the movement of atoms and molecules that are continuously moving around, hitting each other and other objects. This motion is faster for the molecules with a largeramount of energy than the molecules with a smaller amount of energy that causes the former to have more heat. Transfer of heat continues until both objects attain the same temperature or the same speed. This transfer of heat depends upon the nature of the material property determined by a parameter known as thermal conductivity or coefficient of thermal conduction. This parameter helps us to understand the concept of transfer of thermal energy from a hotter to a colder body, to differentiate various objects in terms of the thermal property, and to determine the amount of heat conducted from the hotter to the colder region of an object. The transfer of thermal energy occurs in several situations:

• When there exists a difference in temperature between an object and its surroundings,

• When there exists a difference in temperature between two objects in contact with each other, and

• When there exists a temperature gradient within the same object.

Learning Outcomes

After reading this chapter, the reader will be able to

• Express the meaning of sphere of influence and collision frequency

• Derive the distribution function for the free paths among the molecules and demonstrate the concept of mean free path

• Calculate the expression for mean free path following Clausius and Maxwell

• Derive the expression for pressure exerted by a gas using the survival equation

• Calculate the expressions for viscosity, thermal conductivity, and diffusion coefficient of a gaseous system

• Demonstrate Brownian motion with its characteristics and calculate the mean square displacement of a particle executing Brownian motion

• State the idea of a random walk problem

• Solve numerical problems and multiple choice questions on the mean free path, viscosity, thermal conduction, diffusion, Brownian motion, and random walk

4.1 Introduction

The molecules of an ideal gas are considered as randomly moving point particles. From the concept of kinetic theory of gases (KTG), it is well established that even at room temperature, such point molecules of the ideal gas move at very large speeds. The average value of this speed can be determined assuming that the molecules obey Maxwell's speed distribution law and is given by the following expression

Learning Outcomes

After reading this chapter, the reader will be able to

• State the assumptions of kinetic theory of gases (KTG)

• Explain the concept of pressure and calculate the expression for it

• Demonstrate mathematically the gas laws using the expression for pressure derived from KTG

• Present the kinetic interpretation of temperature

• Derive the expression for specific heat at constant volume 𝐶𝑉 and constant pressure 𝐶𝑃

• Explain the concept of degree of freedom

• Solve numerical problems and multiple choice questions on KTG

2.1 Introduction

The kinetic theory of gases (KTG) is a theoretical model that describes the physical properties of a gaseous system in terms of a large number of submicroscopic particles, such as atoms, molecules, and small particles. These constituent elements are in random motion and collide constantly with each other and also with the walls of the container. Considering the molecular composition and characteristic features of such random motion of the molecules, various macroscopic properties of the gaseous system, such as pressure, temperature, viscosity, thermal conductivity, and mass diffusivity can be explained with the help of KTG. In this theory, it is postulated that the pressure exerted by a gas is due to the collision of atoms or molecules moving at different velocities on the walls of a container. It basically attempts to explain the macroscopic properties that are related to the microscopic phenomenon. The physical properties of solids and liquids, in general, are described by their shape, size, mass, volume, etc. Gases, however, have no definite shape, and size. Furthermore, their mass and volume are not directly measurable. In such cases, the KTG can be successfully applied to extract the physical properties of the gaseous system.

Learning Outcomes After reading this chapter, the reader will be able to

• Know various types of thermodynamic systems such as open, closed, and isolated, and the surroundings

• Classify between intensive and extensive thermodynamic variables

• Understand various types of equilibrium conditions satisfied by a thermodynamic system

• State the zeroth law of thermodynamics and highlight its physical significance

• Comprehend the idea of temperature from the zeroth law of thermodynamics

• Solve numerical problems and multiple choice questions on thermodynamic equilibrium and the zeroth law of thermodynamics

7.1 Introduction

Heat is a form of energy. It can be transformed from one form to another as well as can be transferred between various objects maintained at suitable temperatures. For example, in an electric motor, heat is transformed into mechanical energy by the turbine to power the motor. This mechanical energy is then transformed into electrical energy by the engine to illuminate light bulbs. “Thermodynamics” is a branch of physics that deals with heat and the transformation of heat from one form to another, work, temperature, and their relation to energy, entropy, and other physical properties of matter and radiation. It establishes the relation between heat and various forms of energy and describes the transformations that occur in thermal energy from one energy state to another and how this transformation affects matter. A thermodynamic system is described within a framework based on the four laws of thermodynamics that facilitate a quantitative description of the average macroscopic properties of the system in equilibrium. Macroscopic matter refers to large objects that consist of many atoms and molecules. The average properties of such macroscopic systems are determined by the physical quantities such as volume, pressure, and temperature that do not depend upon the detailed microscopic positions and velocities of the atoms and the molecules comprising the macroscopic system. In the equilibrium state of a thermodynamic system, these average properties also do not change with time. These physical quantities are called thermodynamic coordinates, variables, or parameters. If a subset of these properties are experimentally measured, the rest of them can be calculated using thermodynamic relations. Thermodynamics not only gives the exact description of the state of equilibrium but also provides an approximate description (to a very high degree of precision!) of relatively slow processes. This branch of physics can be successfully applied to a wide variety of topics in science, such as physics, physical chemistry, biochemistry, chemical engineering, and mechanical engineering, but also in other complex fields, such as meteorology.

Learning Outcomes

After reading this chapter, the reader will be able to

• Learn the basic concept of the theory of probability

• List the assumptions used in the derivation of Maxwell's speed distribution law

• Derive Maxwell's speed distribution law and test its validity experimentally

• Calculate average, root mean square and most probable speed, energy, and momentum in one, two, and three dimensions, respectively

• State and prove the law of equipartition of energy

• Calculate the specific heat of gases

• Solve numerical problems and multiple choice questions on the distribution of molecular speed, energy, and momentum

3.1 Introduction

In Chapter 2, various characteristic features of a gaseous system based on the model of the kinetic theory of gases (KTG) have been discussed elaborately. Macroscopic properties and various relations among the thermodynamic variables have been explained in terms of this kinetic model.

According to the assumptions used in this model, a gaseous system is composed of a large number of particles (atoms or molecules) with practically no volume occupied by them. Most of the times, these molecules move randomly through empty space at temperatures above absolute zero, and such motions remain unaffected by the presence of other particles. This motion of the molecules is extremely chaotic and is characterized by straight-line trajectories interrupted by collisions with other molecules or with a physical boundary. In such a collision, the transfer of kinetic energy with a change in direction takes place depending on the nature of the relative kinetic energies of the particles. Any individual molecule collides with others at a huge rate, typically of the order of a billion times per second. This chapter is focused to present a comprehensive and quantitative discussion on the distributions of velocities, energies, and momenta of these molecules in various dimensions.

Measurement of the velocities of the molecules at a given time leads to a large distribution of values; some molecules may move very slowly and others very quickly. As these molecules move constantly in different directions, the velocity could be momentarily equal to zero

Learning Outcomes

After reading this chapter, the reader will be able to

• Gather knowledge about state function and its properties

• Understand the meaning of internal energy and its significance in formulating the first law of thermodynamics

• Formulate the first law of thermodynamics and apply it to various thermodynamic processes

• Grasp the idea of various thermodynamic processes and the related work done in these processes

• Find a relation between specific heats at constant volume and constant pressure for ideal and real gases

• Find expression for isothermal compressibility and volume expansion coefficient for ideal and real gases

• Understand how the temperature of air varies with height assuming an adiabatic process

• Solve numerical problems and multiple choice questions on the first law of Thermodynamics

8.1 Introduction

Systems, surroundings, and interactions between them play a vital role in the development of the subject—thermodynamics. To extract out the physical properties of a thermodynamic system, it is essential to have knowledge about the fundamental laws and concepts of thermodynamics. For example, heat and work are two interrelated concepts. Heat is the transfer of thermal energy between two bodies that are at different temperatures and are not equal to thermal energy. Work is the external physical parameter used to transfer energy between a system and its surroundings. Further, work is needed to create heat and to transfer the thermal energy. Thus, work and heat together allow systems to exchange energy. The relationship between these two physical quantities, heat and work, can be analyzed through the laws and concepts of thermodynamics. The interaction between heat and other types of energy is primarily focused on the topic of thermodynamics. To understand the relationship between heat and work, it is required to have an idea about a third linking factor, known as the change in internal energy

7A Planck's units

In 1899, Max Planck first proposed his radical theory of energy quantization. He proposed to build a system of “natural units” from a few of the more important constants in physics. These important constants include the speed of light, the universal gravitational constant, the Planck constant, and the Boltzmann constant. These quantities have great significance in various fields in physics. Combining these fundamental constant quantities, Planck generated various expressions with units of mass, length, time, and temperature. These units are known as Planck units. These four units signify us something different about the nature of reality. Before describing these units, a brief description about the four fundamental physical constants is presented below.

7A.1 The speed of light 𝑐

The speed of light c in a vacuum is a natural unit for speed and has magnitude 𝑐 = 299, 792, 458 ms−1. According to the special theory of relativity, this speed is the upper limit for the speed at which conventional matter or energy can travel through space. It is the universal “speed limit”. It was recognized during the information age that the photons of electromagnetic radiation and material objects are used as the carriers of information. The speed of light is then a restriction on the speed at which information may travel. The speed of light can be used in time of flight measurements to measure large distances to extremely high precision. In a paper published in 1865, James Clerk Maxwell proposed that light is an electromagnetic wave and travels at speed c. In 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame of reference is a constant and is independent of the motion of the light source.

7A.2 Universal gravitational constant𝐺

The gravitational constant, denoted by the capital letter G, is an empirical physical constant with magnitude 𝐺 = 6.67428 × 10−11 N m2 kg−2. This universal gravitational constant is involved in the calculation of gravitational effects in Newton's law of universal gravitation and in Einstein's theory of general relativity.

6A Radiation as a photon gas

In the nineteenth century, physicists applied classical electromagnetic theory to explain the experimental results of black body radiation but were unable to provide an adequate explanation. This was a major problem to the physicists as classical theory predicted an infinite amount of energy of the radiation emitted from a black body. This gross disagreement was called the “ultraviolet catastrophe”. Max Planck presented a paper on December 14, 1900, in which he guessed the answer to this problem of black body radiation. This guess marked the very beginning of quantum mechanics.

The electromagnetic radiation inside a black body chamber exists as patterns of standing waves or modes. A single mode is like a standing wave on a guitar string and is characterized by a frequency. Planck made a bold hypothesis that the energy of such a mode is quantized, ithat is, only certain energies of these oscillation modes are allowed. Thus, the energy 𝐸𝑛 of 𝑛th mode is given by 𝐸𝑛 = (𝑛 + 12 ) ℏ𝜔 = (𝑛 + 12 ) ℎ𝜈, where 𝜈 is the oscillation frequency, ℎ is Planck's constant, and 𝑛 is an integer starting with 0. Thus, each oscillation mode can exist in any one of an infinite number of energy states whose energies are equally separated by the energy ℏ𝜔. In the discussion of the properties of the photon gas, for the time being, we shall ignore 12 in the expression for 𝐸𝑛 as it has no effect on the results we seek. Therefore, we take 𝐸𝑛 = 𝑛ℏ𝜔 as the energy of the 𝑛th mode whose (angular) frequency is 𝜔. When the energy of a mode is 𝐸𝑛, we say that there are 𝑛 photons in the mode. Each photon has energy equal to ℏ𝜔. Thus, according to the prescription of Max Planck, a black body radiation chamber consists of a number of photons in various energy states with different amounts of energy.

Learning Outcomes

After reading this chapter, the reader will be able to

• Understand the distribution of energy density in a black body radiation as a function of wavelength and temperature

• Derive classical laws of black body radiation such as Wien distribution law and Rayleigh–Jeans law

• Get an idea about the development of quantum theory of radiation

• Understand Planck's quanta postulates and explain the black body radiation spectrum

• Derive Planck's law of black body radiation

• Verify Planck's law of black body radiation experimentally

• Derive Wien distribution law and Rayleigh–Jeans law and explain ultraviolet catastrophe from Planck's law of black body radiation

• Determine the temperature of cosmic microwave background radiation using Planck's law of black body radiation

• Solve numerical problems and multiple choice questions on black body Radiation

11.1 Introduction

Figure 11.1 The whole electromagnetic spectrum. Thermal radiation ranges in frequency from the shortest infrared rays through the visible-light spectrum to the longest ultraviolet rays.

Radiation emitted from the surface of a heated source is known as thermal radiation. In this process, thermal energy is spread out in all directions in the form of electromagnetic radiation and travels directly to its point of absorption at the speed of light. It does not require an intervening medium for its propagation. The wavelength of thermal radiation ranges from the longest infrared rays through the visible-light spectrum to the shortest ultraviolet rays. Such electromagnetic spectrum is shown in Figure 11.1 as a function of frequency. The distribution of radiant energy with their corresponding intensities within various ranges of wavelengths is governed by the temperature of the emitting surface .

4A Chemical Potential

Very often, the term “chemical potential” is not well understood by the students. After studying thermal physics and statistical mechanics for several times, students are still in a lot of confusion about the meaning of the term “chemical potential”. This quantity is represented by the letter 𝜇. Typically, students learn the definition of 𝜇, its properties, its derivation in some simple cases, and its consequences, and work out numerical problems on it. Still, students ask the question: “What is the chemical potential?” and “What does it actually mean?” Attempts are made in this appendix to clarify the meaning of this physical quantity 𝜇 with some simple examples.

The concept of chemical potential has appeared first in the classical works of J. W. Gibbs. Since then, it has become actually a subtle concept in thermodynamics and statistical mechanics. It is not easy to grasp the meaning and significance of chemical potential 𝜇, like thermodynamic concepts such as temperature 𝑇 , internal energy 𝐸, or even entropy 𝑆. In fact, chemical potential 𝜇 has acquired a reputation as a concept not easy to grasp even for the experienced physicist. Chemical potential was introduced by Gibbs within the context of an extensive exposition on the foundations of statistical mechanics. In his exposition, Gibbs considered a grand canonical ensemble of systems in which the exchange of particles occurs with the surroundings. In this description, the chemical potential 𝜇 appears as a constant required for a necessary closure to the corresponding set of equations. Thus, a fundamental connection with thermodynamics is achieved by observing that the unknown constant 𝜇 is indeed related to standard thermodynamic functions like the Helmholtz free energy 𝐹 = 𝑈 − 𝑇 𝑆 or the Gibbs thermodynamic potential 𝐺 = 𝐹 + 𝑃 𝑉 through their first derivatives. 𝜇, in fact, appeared as a conjugate variable to volume V. 4A.1 Comments about chemical potential

We are familiar with the term potential used in mechanical and electrical system. A capacity factor is associated with each potential term. For example, in a mechanical system, mass is the capacity factor associated with the gravitational potential 𝑔(ℎ2 − ℎ1), where ℎ1 and ℎ2 are the corresponding heights, and the gravitational work done is given by 𝑚𝑔(ℎ2 − ℎ1).

Learning Outcomes

After reading this chapter, the reader will be able to

• Demonstrate the meaning of reversible, irreversible, and quasi-static processes used in thermodynamics

• Explain heat engines, and their efficiency and indicator diagram

• Formulate the second law of thermodynamics and apply it to various thermodynamic processes

• Demonstrate an idea about entropy and its variation in various thermodynamic processes

• State and compare various statements of the second law of thermodynamics

• Elucidate the thermodynamic scale of temperature and its equivalence to the perfect gas scale

• Explain the principle of increase of entropy

• Understand the third law of thermodynamics and explain the significance of unattainability of absolute zero

• Solve numerical problems and multiple choice questions on the second law of thermodynamics

9.1 Introduction

The first law of thermodynamics states that only those processes can occur in nature in which the law of conservation of energy holds good. But our daily experience shows that this cannot be the only restriction imposed by nature, because there are many possible thermodynamic processes that conserve energy but do not occur in nature. For example, when two objects are in thermal contact with each other, the heat never flows from the colder object to the warmer one, even though this is not forbidden by the first law of thermodynamics. This simple example indicates that there are some other basic principles in thermodynamics that must be responsible for controlling the behavior of natural processes. One such basic principle is contained in the formulation of the second law of thermodynamics.

This principle limits the use of energy within a source and elucidates that energy cannot be arbitrarily passed from one object to another, just as heat cannot be transferred from a colder object to a hotter one without doing any external work. Similarly, cream cannot be separated from coffee without a chemical process that changes the physical characteristics of the system or its surroundings. Further, the internal energy stored in the air cannot be used to propel a car, or the energy of the ocean cannot be used to run a ship without disturbing something (surroundings) around that object.

1A Calculation of the number of accessible states to an ideal gas

We consider an ideal gas enclosed in a container of volume 𝑉 at a temperature 𝑇. The gas consists of 𝑁 number of molecules, each of mass 𝑚. Suppose the total energy of the system lies in a narrow range from (𝐸 − 𝑑𝐸) to 𝐸. Any molecule of the ideal gas lying within this energy range is described by a state having an elementary volume

where 𝑞′𝑠 and 𝑝′𝑠 are, respectively, the position and momentum coordinates of the molecules of the gaseous system.